In a large Dutch oven or stockpot, heat 3 T of the olive oil over medium-low heat. Add the onion and celery with a sprinkle of salt and pepper, and cook, stirring, until translucent, about 3 minutes. Increase heat to medium high, and add mushrooms. Cook, stirring, for 3 minutes, then add the spinach with a sprinkle of salt and pepper. Cook until spinach is wilted and mushrooms are tender. Remove the vegetables from the pot and reserve.

Add the butter to the same pot, and adjust the heat to medium. Season the chicken on all sides with salt and pepper, and add to the melted butter. Cook, stirring, until the chicken is almost cooked through but not browned, about 6 minutes. Add the wine and simmer until almost completely reduced. Sprinkle the flour over the chicken, and cook, stirring, for about 5 minutes, allowing the flour to coat the chicken and absorb the liquid. Gradually stir in the cream and milk, scraping the pot to release any cooked flour from the bottom and sides (if necessary, stir with a whisk to break up any lumps). Simmer for 5 minutes, stirring constantly, allowing the liquid to thicken. With a spoon or ladle, reserve 1 cup of liquid. Remove the pot from the heat, and add the cooked vegetables and mushrooms. Add the chopped parsley. Season with nutmeg, salt, and pepper to taste.

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To Assemble the Lasagna:

Bring a large pot of salted water to a boil and cook the noodles for about 8 minutes; they should be not quite cooked through (al dente). Strain the noodles in a colander, and rinse in cold water. Drain, then toss noodles with 2 T of olive oil to prevent sticking. Center a rack in the oven and preheat it to 350°F. Butter a 9" x 13" baking pan. Place a layer of 4 noodles on the bottom, overlapping them slightly. Top with one third of the chicken mixture, then one third of the diced fontina. Repeat layering twice, finishing with a layer of noodles. Spread the reserved sauce on the noodles and then sprinkle with Parmesan. (At this stage, the lasagna can be refrigerated overnight. Cover tightly with aluminum foil or plastic wrap.) Cover with aluminum foil or a lid and bake for 30 minutes. Increase heat to 400°F, remove the foil, and continue to bake until golden brown, 10 to 15 minutes more.